Question:
If $\alpha$ and $\beta$ are the zeroes of a polynomial $2 x^{2}+7 x+5$, write the value of $\alpha+\beta+\alpha \beta$.
Solution:
By using the relationship between the zeros of the quadratic ploynomial.
We have,
Sum of zeroes $=\frac{-(\text { coefficient of } x)}{\text { coefficent of } x^{2}}$ and Product of zeroes $=\frac{\text { constant term }}{\text { coefficent of } x^{2}}$
$\therefore \alpha+\beta=\frac{-7}{2}$ and $\alpha \beta=\frac{5}{2}$
Now, $\alpha+\beta+\alpha \beta=\frac{-7}{2}+\frac{5}{2}=-1$
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